Mesh Analysis: Solving for Current in Multiloop Circuits

[dwn]

Homework Statement

Attached an image.

Homework Equations

KCL

The Attempt at a Solution

80 = 10i₁ + 20(i₁ - i₂) + 30(i₁ - i₃)
30 = 12(i₂ - i₃) + 20(i₂ - i₁)
0 = 40i₃ +30(i₃ - i₁) + 12(i₃ - i₂)

80 = 60i₁ - 20i₂ - 30i₃
30 = 20i₁ + 32i₂ - 12i₃
0 = -30i₁ - 12i₂ + 82i₃

i₁ = 3.550 A
i₂ = 1.863 A

I'm really unsure whether I'm completing equations like this correctly...I think I get the general idea, but feel like I'm missing something--like I'm still guessing at all of this. Any help is appreciated. Thanks

 

[NascentOxygen]

You set up the equations correctly, but in the process relabelled the diagram with a different i₂. So once you have solved your equations, you then have to determine values for the original diagram's currents. I haven't checked your working, but is that what you've done? Determined i₁ and i₁ - i₃ ?

 

[asdf12312]

yes he is right. in the image, i2 is actually i1-i3 from your calculations (which is separate from what you used i2 as). so your answer will be i1=i1 and i2=i1-i3. where i2 is different from the i2 that you used. you can call it i2_new if you want.

 

[dwn]

that's actually the part I am a bit confused about. What determines the currents i1 and i2 that we are asked to find? Are they synonymous with the current loops i1, i2, i3? If so, how are they different?

*Edit*

Why is i2 = i1 - i3?

 

[asdf12312]

The wording in the problem was a bit awkward and if anything you can blame that. Normally you would set up circuits in loops in order, so i1 i2 i3 i4 etc. but the i2 that was asked in this problem, you can call it i4 if you want it doesn't really matter.

so this current 'i4', is simply from your equations the current that is shared by both those meshes. since its direction is going down in the same direction as i1, it will be i1 minus (for the loop current going in the opposite direction) i3. so if this i4 was going on the opposite direction (up) it would be i3-i1. make sense?

 

[NascentOxygen]

On the diagram, you are asked to determine the current (i.e., nett current or total current) in the 30 Ohm resistor. With your mesh currents, that will be i1 - i3.

 

[dwn]

Yes, it sort of makes sense. So is it unnecessary to solve for the loops i2 and i3, or are they relevant to solve for, as you say, i4? Also, when solving for i1 (or i5), is there a special equation for this as well?

Or do I use the loops currents i1,i2,i3 to determine i4 and i5? (1-i3 = i4)

 

[asdf12312]

I think I am confusing you. You don't need an 'i5' value because the i1 asked in the question is the same as the i1 you set up in your equation. so the value you found (i1=3.55) assuming its right is the same as i1 asked for. because the current i1 asked for isn't shared with any other mesh, its just i1. if there was a mesh to the other side of i1 (meaning it was between two meshes) it would be the loop with the current going in the same direction as arrow (i1 in this case) minus the loop current (ix) going in opposite direction.

You also need to include all meshes (i2 and also i3) in your equations. while you don't need to solve for i2 really (doesn't ask for it) you still need to include that loop because it changes the values of i1 and i3. You can check this by taking out the loop with i2 and seeing how your answer changes.